A STUDY OF INFLUENZA VIRION USING
FRACTAL BASED GRAPH THEORY
A THESIS REPORT
Submitted by
P.MAGESH BABU
Under the guidance of
Dr. S. SRINIVASAN
in partial fulfillment for the award of the degree of
MASTER OF PHILOSOPHY in
MATHEMATICS
B.S.ABDUR RAHMAN UNIVERSITY (B.S. ABDUR RAHMAN INSTITUTE OF SCIENCE & TECHNOLOGY)
(Estd. u/s 3 of the UGC Act. 1956) www.bsauniv.ac.in
May 2011
B.S.ABDUR RAHMAN UNIVERSITY (B.S. ABDUR RAHMAN INSTITUTE OF SCIENCE & TECHNOLOGY)
(Estd. u/s 3 of the UGC Act. 1956) www.bsauniv.ac.in
BONAFIDE CERTIFICATE
Certified that this thesis report on A STUDY OF INFLUENZA VIRION
USING FRACTAL BASED GRAPH THEORY is the bonafide work of
P.MAGESH BABU (RRN: 1040201) who carried out the thesis work under
my supervision. Certified further, that to the best of my knowledge the work
reported herein does not form part of any other thesis report or dissertation
on the basis of which a degree or award was conferred on an earlier
occasion on this or any other candidate.
SIGNATURE SIGNATURE
Dr. S. SRINIVASAN Dr. S. SRINIVASAN
SUPERVISOR HEAD OF THE DEPARTMENT
Professor Professor & Head
Department of Mathematics Department of Mathematics
B.S. Abdur Rahman University B.S. Abdur Rahman University
Vandalur, Chennai – 600 048 Vandalur, Chennai – 600 048
TABLE OF CONTENTS
CHAPTER NO. TITLE PAGE NO.
ABSTRACT v
LIST OF FIGURES vi
LIST OF SYMBOLS AND ABBREVATIONS vii
1. INTRODUCTION
1.1 GRAPH THEORY 1
1.1.1 Definition of Graph 2
1.1.2 Loop and Multiple Edges 2
1.1.3 Simple Graph 3
1.1.4 Connected Graph 3
1.1.5 SubGraph 4
1.2 GRAPH COLOURING 5
1.2.1 Edge Colouring 7
1.3 APPLICATIONS 8
1.3.1 Vertex Colouring Applications 9
1.3.2 Edge Colouring Applications 10
1.4 FRACTAL 11
1.4.1 THE METHODS FOR CALCULATING
THE DIMENSION 11
1. 4.1.1 Box Counting Method 11
1. 4.1.2 Cluster Growing Method 11
1.4.2 THE LUNGS 12
2. LITERATURE REVIEW 13
3. INFLUENZA VIRION
3.1 INTRODUCTION 22
3.2 TRANSMISSION 23
3.3 REPLICATION 24
4. ANALYSIS 25
5. CONCLUSION 28
6. REFERENCES 29
7. PAPER PROCEEDINGS 31
8. TECHNICAL BIOGRAPHY 34
ABSTRACT
The present study focuses on the Replication Stages of Influenza
Virion in the Human body (Influenza disease caused by RNA Viruses),
represented by using Graph Theory (Vertex Colouring) has been made. The
Virion can only replicate in the living cells. Influenza Replication is a multi-
step Process. This multi-step process has been explained through Graphical
representation and Graph Colouring (Vertex colouring) has been used to
show the different stages occurring during Replication Process.
In addition, the infected parts of the cell are studied by using Fractal
Dimension Method. There are two methods for Calculating Fractal
Dimension. One is Box Counting Method and second is Cluster Growing
Method. The human lungs are made up of Fractal, so to make the study of
infection on the lungs, the Fractal dimension is chosen. In the above two
methods, the cluster growing method is more perfect to study the infected
parts of the cells. It is used to predict the growth of the infection as well.
v
LIST OF FIGURES
Figure Number
Graph of Konigsberg Bridge Problem 1.1
Graph of Loop and Multiple Edges 1.2
Simple Graph 1.3
Connected Non-Simple Graph 1.4
Disconnected Simple Graph 1.5
Connected Graph 1.6
SUB GRAPHS 1.7
Graph Colouring
3-Colouring 1.8
4-Colouring 1.9
5-Colouring 1.10
Not a Permissible Colouring 1.11
4- Edge Colouring 1.12
5- Edge Colouring 1.13
6- Edge Colouring 1.14
Not a Permissible Colouring 1.15
Structure of Influenza Virion 3.1
Replication of Virion (Graph Colouring) 4.1
Structure of Infected Lungs 4.2
vi
LIST OF SYMBOLS AND ABBREVIATIONS
- Average mass of the clusters
df - Dimension
E (G) - Edge Set
G - Graph
IG - Incidence map
l - Minimum distance
-VRNA - Negative-sense viral Ribonucleic acid
+VRNA - Positive-sense viral Ribonucleic acid
- Summation
X - Sum of the VRNA
ni - Number of VRNA , i = 1 to n.
V(G) - Vertex Set
VRNA - Viral Ribonucleic acid
HA - Hemagglutinin
NA - Neuraminidase
M1 - Matrix 1
M2 - Matrix 2
NP - Nucleoprotein
NSP1 - Non-structural protein 1
vii
NSP2 - Non-structural protein 2
NEP - Nuclear Export Protein
PA - Polymerase acidic protein
PB1 - Polymerase basic protein 1
PB2 - Polymerase basic protein 2
PB1-F2 - Polymerase basic protein 1 – F2
vRNPs - viral ribonucleoproteins
Pol II - Cellular RNA Polymerase II
CTD - C-terminal repeat domain
snRNAs - small nuclear RNAs
NS1-BP - NS1 binding protein
CPSF - Cleavage and polyadenylation specificity factor
PABPII - Poly(A) binding protein II
RdRp - RNA dependent RNA polymerase
mRNAs - messenger RNAs
χ(G) - Chromatic number of G
χ’(G) - Chromatic index of G
viii
1. INTRODUCTION
1.1 GRAPH THEORY
Graph theory is a Mathematical model for most of the system.
In general, graphs have an intuitive and aesthetic appeal because of their
diagrammatic representation. Graph theory has already found its
applications in almost all fields of study such as Physics, Chemistry,
Communication Science, Electrical net work analysis, Operations Research,
Genetics etc. The theory is intimately linked to many branches of
mathematics like, Matrix Theory, Group Theory, Probability, Topology and
Combinatorics.
Euler (1736) was one of the earliest workers on this subject. In fact,
he became the father of graph theory (as well as topology) when he settled
the famous unsolved problem of his day called the Konigsberg bridge
problem in the negative. The problem is to begin at any one of the four land
areas, walk across each bridge exactly once and return to the starting point.
In attacking the problem Euler replaced each land area by a point and each
bridge by a line joining the corresponding points, there by producing a graph.
Figure 1.1
1
Informally, a graph is a diagram consisting of points, called vertices,
joined together by lines, called edges; each edge joins exactly two vertices.
A graph G is consisting of a vertex set V(G), an edge set X(G), and a relation
that associates with each edge two vertices (not necessarily distinct) called
its endpoints.
1.1.1 Definition of Graph
A graph G consists of a pair (V(G),X(G)) where V(G) is a non-empty
finite set whose elements are called points or vertices and X(G) is a set of
unordered pairs of distinct elements of V(G). The elements of X(G) are called
lines or edges of the graph G. If x = {u,v}∈X(G), the line x is said to join u
and v. We write x = uv and we say that the points u and v are adjacent.
We also say that the point u and the line x are incident with each other.
If two distinct lines x and y are incident with a common point then they
are called adjacent lines. A graph with p points and q lines is called a (p,q)
graph.
1.1.2 Loop and Multiple Edges
A loop is an edge whose endpoints are equal i.e., an edge joining a
vertex itself is called a loop. We say that the graph has multiple edges if in
the graph two or more edges join the same pair of vertices.
Figure 1.2
2
1.1.3 Simple Graph
A graph with no loops or multiple edges is called a simple graph. We
specify a simple graph by its set of vertices and set of edges, treating the
edge set as a set of unordered pairs of vertices and write e = uv (or e = vu)
for an edge e with endpoints u and v.
Simple Graph
Figure 1.3
1.1.4 Connected Graph
A graph that is in one piece is said to be connected, whereas one
which splits into several pieces is disconnected.
A graph G is connected if there is a path in G between any given pair
of vertices, otherwise it is disconnected. Every disconnected graph can be
split up into a number of connected subgraphs, called components.
Connected Non-Simple Graph
Figure 1.4
3
Disconnected Simple Graph
Figure 1.5
1.1.5 Subgraph
Let G be a graph with vertex set V(G) and edge-list E(G). A sub graph
of G is a graph all of whose vertices belong to V(G) and all of whose edges
belong to E(G). For example, if G is the connected graph below:
Figure1.6
where V(G) = {u, v, w, z} and E(G) = (uv, uw, vv, vw, wz, wz} then the
following four graphs are subgraphs of G.
4
Figure 1.7
1.2 GRAPH COLOURING
In graph theory, graph colouring is a special case of graph labeling; it
is an assignment of labels traditionally called "colours" to elements of a graph
subject to certain constraints. Let G be a graph with no loops. A k-colouring
of G is an assignment of k colours to the vertices of G in such a way that
adjacent vertices are assigned different colours. If G has a k-colouring, or G
is k-colouring, then G is said to be k-colourable. The chromatic number of G,
denoted by X(G), is the smallest number k for which is k-colourable.
For example,
5
3-colouring Figure 1.8
4-colouring Figure 1.9
5-colouring Figure 1.10
Figure 1.11
6
In Figure 1.11, Graph is not a permissible colouring, since one of the edge
has colour blue at both ends.
It is easy to see from above examples that chromatic number of G is
at least 3. That is X(G) ≤ 3, since G has a 3-colouring in first diagram. On the
other hand, X(G) ≥ 3, since G contains three mutually adjacent vertices
(forming a triangle)., which must be assigned different colours. Therefore, we
have X(G) = 3.
1.2.1 Edge Colourings
Let G be a graph with no loops. A k-edge-colouring of G is an
assignment of k colours to the edges of G in such a way that any two edges
meeting at a common vertex are assigned different colours. If G has a k-edge
colouring, then G is said to be k-edge colourable. The chromatic index of G,
denoted by X`(G), is the smallest k for which G is k-edge-colourable.
For example, consider the following graphs with eight edges.
4-edges-colouring Figure 1.12
5-edge-colouring Figure1.13
7
6-edge-colouring Figure 1.14
Figure 1.15
In Figure 1.15, Graph is not a permissible colouring. Since two of the edges
coloured blue meet at a common vertex.
From the above examples, it follows that X`(G) ≤ 4, since G has a
4-edge-colouring in figure a (above). On the other hand, X`(G) ≥ 4, since G
contains 4 edges meeting at a common vertex i.e., a vertex of degree 4,
which must be assigned different colours. Therefore, X` (G) = 4.
1.3 APPLICATIONS
Graphs are among the most ubiquitous models of both natural and
human-made structures. They can be used to model many types of relations
and process dynamics in physical, biological and social systems. Many
problems of practical interest can be represented by graphs.
In computer science, graphs are used to represent networks of
communication, data organization, computational devices, the flow of
computation, etc.
8
Graph theory is also used to study molecules
in chemistry and physics. In condensed matter physics, the three
dimensional structure of complicated simulated atomic structures can be
studied quantitatively by gathering statistics on graph-theoretic properties
related to the topology of the atoms.
Graph theory is also widely used in sociology as a way, for example,
to measure or to explore diffusion mechanisms, notably through the use
of social network analysis software.
Graph theory is useful in biology and conservation efforts where a
vertex can represent regions where certain species exist (or habitats) and the
edges represent migration paths, or movement between the regions. This
information is important when looking at breeding patterns or tracking the
spread of disease, parasites or how changes to the movement can affect
other species.
Graph colouring is a practical method of representing many real world
problems including time scheduling, frequency assignment, register
allocation, and circuit board testing.
1.3.1 Vertex colouring applications
Many scheduling problems involve allowing for a number of pairwise
restrictions on which jobs can be done simultaneously. For instance,
in attempting to schedule classes at a university, two courses taught by the
same faculty member cannot be scheduled for the same time slot. Similarly,
two courses that are required by the same group of students also should not
conflict. The problem of determining the minimum number of time slots
needed subject to these restrictions is a graph colouring problem. The time
slots are colours for the vertices, the vertices are courses, and the edges
between courses are restrictions that force different time slots.
9
One very active application for vertex colouring is register allocation.
The register allocation problem is to assign variables to a limited number of
hardware registers during program execution. We want to put as many
variables in registers as possible for quick execution of the program. There
are typically far more variables than registers so it is necessary to assign
multiple variables to registers. Variables conflict with each other if one is
used both before and after the other within a short period of time. The goal is
to assign variables that do not conflict so as to minimize the use of
non-register memory.
1.3.2 Edge colouring applications
Let us assume that we need to schedule a given set of two-person
interviews, where each interview takes one hour. All meetings could be
scheduled to occur at distinct times to avoid conflicts, but it is less wise to
schedule nonconflicting events simultaneously. We can construct a graph
whose vertices are the people and whose edges represent the pairs of
people who want to meet.
An edge colouring of this graph defines the schedule. The colour
classes represent the different time periods in the schedule, with all meetings
of the same colour happening simultaneously.
The National Football League solves such an edge colouring problem
each season to make up its schedule. Each team's opponents are
determined by the records of the previous season. Assigning the opponents
to weeks of the season is the edge-colouring problem. Edge colouring can
be reduced to vertex colouring (in linear time) by constructing the line graph
of the input graph G. This is the graph constructed by replacing each edge
with a vertex, and connected vertices in the new graph according to the
edges that share a vertex in the original graph.
10
1.4 FRACTAL
A fractal is “a rough or fragmented Geometric shape that can be split
into parts, each of which is a reduced size copy of the whole; a property
called self – similarity.
1.4.1 THE METHODS FOR CALCULATING THE DIMENSION
There are two methods for calculating Dimension. They are
Box Counting Method
Cluster Growing Method
1.4.1.1 Box Counting Method
Let NB be the number of boxes of linear size 𝒍B, needed to cover the
given network. The fractal dimension dB is then given by
NB ~ 𝒍 −𝒅
This means that the average number of vertices 𝑴 𝒍 within a box of
size 𝒍B 𝑴 𝒍 ~𝒍𝒅
By measuring the distribution of N for different box sizes or by measuring the
distribution of 𝑴 𝒍 for different box sizes, the fractal dimension 𝒅 can
be obtained by a power law fit of the distribution.
1.4.1.2 Cluster Growing Method
One seed node is chosen randomly. If the minimum distance 𝒍 is
given, a cluster of nodes separated by at most 𝒍 from the seed node can be
formed. The procedure is repeated by choosing many seeds until the
clusters cover the whole network.
11
Then the dimension 𝐝𝐟 can be calculated by 𝑴 ~ 𝒍𝒅𝒇
Where 𝑴 is the average mass of the clusters, defined as the average
number of nodes in cluster.
1.4.2 THE LUNGS
The first place where this is found is rather obvious to anyone who
knows fractals — in the pulmonary system, which we use to breathe. The
pulmonary system is composed of tubes, through which the air passes into
microscopic sacks called alveoli. The main tube of the system is trachea,
which splits into two smaller tubes that lead to different lungs, called the
bronchi. The bronchi are in turn split into smaller tubes, which are even
further split. This splitting continues further and further until the smallest
tubes, called the bronchioles lead into the alveoli.
12
2. LITERATURE REVIEW
Tasleem Samji (2009) examined the Influenza A: Understanding the
viral life cycle. They found that the influenza virus life cycle can be divided
into the following stages: entry into the host cell; entry of vRNPs into the
nucleus; transcription and replication of the viral genome; export of the
vRNPs from the nucleus; and assembly and budding at the host cell plasma
membrane.
HA is a homotrimer that forms spikes on the viral lipid membrane. The
influenza virus, for instance, has two types of spikes, one type, composed of
hemagglutinin protein (HA), fuses with the host cell membrane, allowing the
virus partial to enter the cell. The other type of spike, composed of the
protein neuraminidase (NA) helps the newly formed virus particles to bud out
from the host cell membrane.
Influenza viral transcription and replication occurs in the nucleus;
therefore, after being released into the cytoplasm, the VRNP should enter the
Nucleus. The influenza viral genome is made up of negative sense strands
of RNA. In order for the genome to be transcribed, it first should be
converted into a positive sense RNA to serve as a template for the
production of viral RNAs.
Replication of the genome does not require a primer; instead, the viral
RNA dependent RNA polymerase initiates RNA synthesis internally on viral
RNA.
It is known that only negative sense vRNPs are exported from the
Nucleus. After the vRNPs have left the nucleus all that is left for the virus to
do is to form viral particles and leave the cell. It uses the host cell’s plasma
membrane to form the viral particles that leave the cell and go onto infect
neighboring cells. Virus particles bud from the apical side of polarized cells.
13
One of the most important steps that should occur before the newly
made viral particles leave the plasma membrane is the cleavage of sialic acid
residue from glycoproteins and glycolipids. NA removes these sialic acids.
Without this process, the viral particle would not be released from the plasma
membrane. This is just a brief overview of the steps in the Influenza A viral
life Cycle from entry into a host cell to exit from the host cell.
Hampson AW, Mackenzie JS (2006) examined the Influenza viruses
and According to them the human epidemic influenza is caused by influenza
type A and B viruses, which continually undergo antigenic change in their
surface antigens, haemagglutinin (H) and neuraminidase (N).
Influenza epidemics is the consequence of small, ongoing antigenic
changes known as “antigenic drift”, which occurs in both influenza types.
Pandemic influenza occurs at irregular and unpredictable intervals, and is
the result of a major antigenic change known as “antigenic shift”, which
occurs only in influenza A.
Aquatic birds are the evolutionary hosts of influenza viruses; they
harbour many distinct forms or subtypes of influenza A, which are usually
present as harmless gut infections. Antigenic shift involves the evolution of a
new human influenza A virus through the acquisition of a new
haemagglutinin gene encoding a different subtype from an avian influenza,
or by the adaptation of an avian virus, causing it to become transmissible
between humans.
Two subtypes of avian influenza, H5 and H7, can cause severe
infections when introduced into domestic poultry. Recently, influenza
A/H5N1 viruses have caused widespread outbreaks, starting in Asia and
spreading widely to other regions. Avian influenza viruses do not readily
infect humans. However, during the past 3 years, more than 250 cases of
H5N1 infection of humans have occurred, with associated mortality
approaching 60%. It is feared that a new pandemic of human influenza may
emerge from this.
14
Kawaguchi A, Nagata K.(2006) discuss that the genome of influenza
A virus is a set of eight segmented- and single-stranded RNAs. A basic
transcription and replication unit is the genome complexed with viral RNA-
dependent RNA polymerases and nucleoprotein (NP). For the efficient
transcription and replication of the genome, not only viral factors but also
host cell-derived factors are required.
Although receptor and protease molecules play important roles in
infection and pathogenicity, it is also possible that host factors involved in the
virus genome function determine these. PB2, for instance, is reported to be a
possible candidate for determination of the host range of avian influenza
viruses. Here we summarize recent progresses in the molecular mechanism
of the influenza virus genome transcription and replication and discuss the
involvement of host factors in these processes.
Kliegman RM, Behrman RE, Jenson HB. Nelson (2007) discuss
some concepts about Influenza The influenza viruses are complex and they
change from year to year in response to mutations and evolutionary
pressure. Three major types of virus (A, B, and C) are known, only two of
which (A and B) cause significant diseases in humans; other types of
influenza viruses are known to infect domestic animals such as birds and
pigs. A typical case of influenza starts with high fever, chills, muscle
aches, headaches, a nonproductive cough, and/or a distinct feeling of
unwellness. These symptoms are soon followed by worsening upper
respiratory symptoms [nasal congestion, running nose, (sore throat, and/or
drenching sweats). Patients may feel profoundly ill and weak.
In most individuals, the illness lasts for about a week and is followed
by complete recovery, with most people thinking that they have suffered from
an especially bad "cold" (a different type of viral syndrome). Due to variations
in patients' response to the disease and because of the large number of
influenza subtypes, the course of the disease is unpredictable and, therefore,
potentially lethal, even in otherwise healthy adults and children.
15
Routine vaccination and diligent surveillance efforts are the most
important ways to control the disease and limit the spread of outbreaks.
Since the viruses are constantly evolving, decisions must be made every
year by public health authorities regarding the viral strains that vaccines will
be targeted against. Even though vaccines are specific for individual strains
of the virus, there is evidence that people in high-risk groups who get yearly
flu shots tend to have milder illnesses when new strains emerge.
The virus can reside in domesic animals and cause disease in
humans, and animal-to-human transmission is facilitated by close interaction
with animals. Prevention efforts should therefore include good animal
husbandry practices. Extensive exposure to domestic birds was one of the
factors that contributed to the avian influenza H5N1 outbreak in Asia in 2006.
The H5N1 strain of influenza A is especially dangerous, with mortality rates in
excess of 50%.
Huawei Mao, Wenwei Tu*, Yinping Liu, Gang Qin, Jian
Zheng,Ping-Lung Chan, Kwok-Tai Lam, J. S. Malik Peiris, and Yu-Lung
Lau (2010) examined the Inhibition of human natural killer cell activity by
influenza virion and hemagglutinin. They found that Natural killer (NK) cells
keep viral infections under control at the early phase by directly killing
infected cells. Influenza is an acute contagious respiratory viral disease
transmitting from host-to-host in the first few days of infection. Evasion of host
innate immune defenses including NK cells would be important for its
success as a viral pathogen of humans and animals
NK cells encounter influenza virus within the micro-environment of
infected cells. It is therefore important to investigate the direct effects of
influenza virus on NK cell activity. Recently they demonstrated that influenza
virus directly infects human NK cells and induces cell apoptosis to counter
their function. Here, they further demonstrated that both the intact
influenza virion and free hemagglutinin protein inhibited the cytotoxicity of
fresh and IL-2-activated primary human NK cells.
16
Hemagglutinin was bound and internalized into NK cells via the sialic
acids. This interaction did not decrease NKp46 expression, but caused
the down-regulation of chain through lysosomal pathway, which in turn
caused the decrease of NK cell cytotoxicity mediated by NKp46 and
NKp30. The underlying dysregulation of signaling pathway involved chain
down-regulation leading to decreased Syk and ERK activation and granule
exocytosis upon target cell stimulation, finally causing reduced cytotoxicity.
These findings suggest that influenza virus may have developed a novel
strategy to evade NK cell innate immune defense which is likely to facilitate
viral transmission and may also contribute to virus pathogenesis.
Clark T, Stephenson I. Influenza A/H1N1 (2009) discuss about the
Influenza viruses. According to them Influenza viruses belong to a
family Orthomyxoviridae, family of single-stranded, negative-sense
RNA viruses causing influenza and other respiratory diseases. Aquatic birds
are the evolutionary hosts and the major reservoir of influenza A viruses.
These zoonotic viruses exhibit great genetic diversity and infect the wide
range of host species including humans. They caused several pandemics
that resulted in unprecedented number of deaths throughout the world.
Liu Y, Lou Z, Bartlam M, Rao Z (2009) discuss that the influenza
virus RNA-dependent RNA polymerase is a heterotrimeric complex (PA, PB1
and PB2) with multiple enzymatic activities for catalyzing viral RNA
transcription and replication. The roles of PB1 and PB2 have been clearly
defined, but PA is less well understood. The critical role of the polymerase
complex in the influenza virus life cycle and high sequence conservation
suggest that it should be a major target for therapeutic intervention.
However, until very recently, functional studies and drug discovery
targeting the influenza polymerase have been hampered by the lack of three-
dimensional structural information. They will review the recent progress in the
structure and function of the PA subunit of influenza polymerase, and discuss
prospects for the development of anti-influenza therapeutics based on
available structures.
17
Stella SF Ng, Olive TW Li, Timothy Kw Cheung, JS Malik Peiris and
Leo LM Poon (2008) found that the initiation of transcription and replication
of influenza A virus required the 5’ and 3’ ends of vRNA. Here, the role of
segment specific non-coding sequences of influenza A virus on viral RNA
synthesis was studied Recombinant viruses, with the nonstructural protein
(NS) segment-specific non-coding sequences replaced by the corresponding
sequences of the neuraminidase (NA) segment, were characterized.
The NS and NA vRNA levels in cells infected with these mutants were
much higher than those of the wild type. Whereas the NS and NA mRNA
levels of the mutants were comparable to the wild-type levels. In contrast,
the PB2 vRNA with heterologous segment-specific non-coding sequences
was not affected by the mutations. The observations suggested that, with the
cooperation between the homologous 5’ and 3’ segment-specific sequences,
the introduced mutations could specifically enhance the replication of NA and
NS vRNA.
According to Nayak dP, Hui EK, Barman S(2004), the Influenza
viruses are causative agents of an acute febrile respiratory disease called
influenza (commonly known as "flu") and belong to the Orthomyxoviridae
family. These viruses possess segmented, negative stranded RNA genomes
(vRNA) and are enveloped, usually spherical and bud from the plasma
membrane (more specifically, the apical plasma membrane of polarized
epithelial cells). Complete virus particles, therefore, are not found inside
infected cells. Virus particles consist of three major subviral components,
namely the viral envelope, matrix protein (M1), and core (viral
ribonucleocapsid [vRNP]).
The viral envelope surrounding the vRNP consists of a lipid bilayer
containing spikes composed of viral glycoproteins (HA, NA, and M2) on the
outer side and M1 on the inner side. Viral lipids, derived from the host plasma
membrane, are selectively enriched in cholesterol and glycosphingolipids.
18
M1 forms the bridge between the viral envelope and the core. The viral core
consists of helical vRNP containing vRNA (minus strand) and NP along with
minor amounts of NEP and polymerase complex (PA, PB1, and PB2). For
viral morphogenesis to occur, all three viral components, namely the viral
envelope (containing lipids and transmembrane proteins), M1, and the vRNP
must be brought to the assembly site, i.e. the apical plasma membrane in
polarized epithelial cells.
Finally, buds must be formed at the assembly site and virus particles
are released with the closure of buds. Transmembrane viral proteins are
transported to the assembly site on the plasma membrane via the exocytic
pathway. Both HA and NA possess apical sorting signals and use lipid rafts
for cell surface transport and apical sorting. These lipid rafts are enriched in
cholesterol, glycosphingolipids and are relatively resistant to neutral
detergent extraction at low temperature.
M1 is synthesized on free cytosolic polyribosomes. vRNPs are made
inside the host nucleus and are exported into the cytoplasm through the
nuclear pore with the help of M1 and NEP. How M1 and vRNPs are directed
to the assembly site on the plasma membrane remains unclear. The likely
possibilities are that they use a piggy-back mechanism on viral glycoproteins
or cytoskeletal elements.
Alternatively, they may possess apical determinants or diffuse to the
assembly site, or a combination of these pathways. Interactions of M1 with
M1, M1 with vRNP, and M1 with HA and NA facilitate concentration of viral
components and exclusion of host proteins from the budding site. M1
interacts with the cytoplasmic tail (CT) and transmembrane domain (TMD) of
glycoproteins, and thereby functions as a bridge between the viral envelope
and vRNP. Lipid rafts function as microdomains for concentrating viral
glycoproteins and may serve as a platform for virus budding. Virus bud
formation requires membrane bending at the budding site.
19
A combination of factors including concentration of and interaction
among viral components, increased viscosity and asymmetry of the lipid
bilayer of the lipid raft as well as pulling and pushing forces of viral and host
components are likely to cause outward curvature of the plasma membrane
at the assembly site leading to bud formation.
Eventually, virus release requires completion of the bud due to fusion
of the opposing membranes, leading to the closure of the bud, separation of
the virus particle from the host plasma membrane and release of the virus
particle into the extracellular environment. Among the viral components, M1
contains an L domain motif and plays a critical role in budding. Bud
completion requires not only viral components but also host components.
However, how host components facilitate bud completion remains unclear.
In addition to bud completion, influenza virus requires NA to release
virus particles from sialic acid residues on the cell surface and spread from
cell to cell. Elucidation of both viral and host factors involved in viral
morphogenesis and budding may lead to the development of drugs
interfering with the steps of viral morphogenesis and in disease progression.
Amorim MJ, Digard P(2006) discuss that over 20 years since the
publication of experiments that showed that influenza A virus RNA synthesis
takes place in the cell nucleus and that here, the virus subverts the cellular
transcription machinery to express and replicate its own single-strand RNA
genome. Since then, our understanding of the organization of the nucleus
has increased enormously, particularly with regard to the functional
integration of the RNA polymerase II transcript some. This review
summarizes the recent progress in defining the intimate association between
the viral and cellular transcriptional machinery.
N Balakrishnan(1991) discusses that the primary objective in
examination scheduling is that no student should have more than one exam
during the same period, most real-life cases have a large number of other
complicating constraints that make the problem complex. These may include
20
room availability constraints, prevention of exams in successive periods for
the same student, incompatibility of a few exams with certain periods, etc.
He describes the successful application of a graph-colouring based
examination scheduling heuristic to the scheduling problem faced by the
Freeman School of Business at Tulane University. The procedure, which is
currently being used at the school, is easy to implement and handles all the
issues in the problem including the room availability constraints
simultaneously. Our computational experience indicates that the procedure
is general and flexible enough to be easily adapted to exam scheduling
problems at other small schools.
In Applications of Graph Coloring, Unal Ufuktepe and Goksen
Bacak (2005) discuss that a graph G is a mathematical structure consisting
of two sets V(G) (vertices of G) and E(G) (edges of G). Proper colouring of a
graph is an assignment of colours either to the vertices of the graphs, or to
the edges, in such a way that adjacent vertices / edges are coloured
differently. This paper discusses colouring and operations on graphs
with Mathematica and webMathematica. they consider many classes of
graphs to colour with applications. They draw any graph and also try to show
whether it has an Eulerian and Hamiltonian cycles by using our package
Colour G.
Baki Koyuncu ,Mahmut Seçir discussed that the Graph Colouring
Algorithm was used to generate the student weekly time table in a typical
university department. The problem was a Node-Point problem and it could
not be solved in the polynomial domain. Various constraints in weekly
scheduling such as lecturer demands, course hours and laboratory
allocations were confronted and weekly time tables were generated for 1st,
2nd, 3rd and 4th year students in a typical semester.
21
3. INFLUENZA
3.1 INFLUENZA VIRION
Influenza, commonly referred to as the flu, is an infectious disease
caused by RNA viruses of the family Orthomyxomiridae (the influenza
viruses). Influenza viruses have single-stranded RNA genomes, consisting
of eight segments of negative polarity. Influenza is described as a viral
infection of the lungs characterized by fever, cough, and severe muscle
aches, that affect birds and mammals.[6][8]
The most common symptoms of the disease are chills, fever, sore
throat, muscle pains, severe headache, coughing, weakness/fatigue and
general discomfort. Although it is often confused with other influenza-like
illnesses, especially the common cold, influenza is a more severe disease
than the common cold and is caused by a different type of virus. Influenza
may produce nausea and vomiting, particularly in children, but these
symptoms are more common in the unrelated gastroenteritis, which is
sometimes called "stomach flu" or "24-hour flu".
Typically, influenza is transmitted through the air by coughs or
sneezes, creating aerosols containing the virus. Influenza can also be
transmitted by direct contact with bird droppings or nasal secretions, or
through contact with contaminated surfaces. Airborne aerosols have been
thought to cause most infections, although which means of transmission is
most important is not absolutely clear.
Influenza viruses can be inactivated by sunlight, disinfectants and
detergents. As the virus can be inactivated by soap, frequent hand washing
reduces the risk of infection as observed by Hampson AW, Mackenzie[6], [8]
22
Figure 3.1Structure of Influenza virion
3.2 TRANSMISSION
Influenza virus shedding (the time during which a person might be
infectious to another person) begins the day before symptoms appear and
the virus is then released for between 5 to 7 days, although some people
may shed virus for longer periods. People who contract influenza are more
infective between the second and third days after infection. The amount of
virus shed appears to correlate with fever, with higher amounts of virus shed
when temperatures are highest. Children are much more infectious than
adults and shed virus from just before they develop symptoms until two
weeks after infection. The transmission of influenza can be modeled
mathematically, which helps predict how the virus will spread in a population.
Influenza can be spread in three main ways: by direct transmission
(when an infected person sneezes mucus directly into the eyes, nose or
mouth of another person); the airborne route (when someone inhales the
aerosols produced by an infected person coughing, sneezing or spitting) and
through hand-to-eye, hand-to-nose, or hand-to-mouth transmission, either
from contaminated surfaces or from direct personal contact such as a hand-
shake. The relative importance of these three modes of transmission is
unclear, and they may all contribute to the spread of the virus.
23
3.3 REPLICATION
Virion can only replicate in the living cells. Influenza infection and
Replication is a multi – step process. Firstly the virus breaks and enters into
the cell, then delivers its genome to a site where it can produce new copies
of viral proteins and RNA, assembles these components into new viral
particles and finally exits the host cell.
Once the virus enters the host cell, the core proteins and the eight
segmented negative sense VRNA are transported into the nucleus. In the
nucleus, the structural proteins are separated from the negative sense VRNA
and get transmitted into the cytoplasm.
Negative senses VRNA are converted to positive senses VRNA due to
the presence of enzymes in the host cell. The positive senses VRNA
produce multi number of copies.
Afterwards, the positive senses VRNA are converted to negative
senses VRNA due to the presence of enzymes in the host cell. In addition,
the structural protein has done their multiplication process in the Ribosome’s.
Then the negative senses VRNA and structural proteins are combined
together and they form a new virion which contains eight segments of
negative VRNA; In the same way multi number of copies are delivered from
the host cell as observed by Kawaguchi A, Nagata K [9].
24
4. ANALYSIS
Graphs serve as Mathematical models to analize successfully many
concrete real world problems. In this paper, the Replication of virion is
represented by the use of Graph theory, especially by using vertex colouring
and labeling. By the application of vertex colouring, the different stages that
occur in Replication of virion are seen from the Figure 4.1. In this Figure,
vertices represent the stages and the edges represent the connectivity of the
different stages.
Replication of virion (Graph Colouring)
Figure 4.1
25
The above graphical representation is about Replication of Influenza
Virion. The vertex which is assigned by Red colour is the Virion which is
composed of eight segmented negative VRNA. The Vertices with Blue
colour denote the negative VRNA which are delivered from the Virion. From
each negative VRNA, the Structural proteins are separated; it is denoted by
an arrow mark towards the cytoplasm. In the cytoplasm n- number of
vertices is coloured by Brown colour, which represents the multiple numbers
of Structural proteins.
The vertices assigned with Bright Green Colour denote the positive
VRNA, which are converted from the Vertices with Blue Colour by the use of
enzymes present in the Nucleus. The vertices with Sea Green Colour
represent the Multiple Numbers of Copies of positive VRNA. Afterwards, the
vertices with Sea Green Colour are converted to Blue Colour due to the
presence of enzymes in the host cell. This ‘n’ – number of vertices with Blue
Colour denotes the ‘n’ – number of negative VRNA, which are the complex of
structural proteins. Afterwards, every eight Blue coloured vertices get
composed and form the Vertex with Red Colour. Therefore multiple copies of
Virion exit from the host cell.
Let, X = Sum of the VRNA.
X =∑ 𝑛𝑖8𝑖=1
Number of Virion = 𝑿𝟖
Where, ni = number of vRNA, i = 1 to n.
The present study explains the Replication of Virion by the use of
Graph Colouring (Vertex Colouring), then Fractal dimension is used for
predicting the growth of the infection.
26
Cluster Growing Method
One virus infected alveoli is chosen randomly. If the minimum
distance 𝒍 is given, a cluster of virus infected alveoli separated by at most 𝒍 from the infected cell which is chosen first. The procedure is repeated by
choosing many virus infected alveoli.
Then the dimension df can be calculated
(from Song.c, Havlin.S, and H. A. Makse [12])
Where is the average mass of the clusters, defined as the average
number of infected alveoli in a cluster. By this way we can predict the growth
of the virus infection in lungs.
Virus Infected Lungs
Figure 4.2
27
5. CONCLUSION
In this paper the Replication of virion is represented by using graph
colouring. Here we can easily see the different stages that occur during the
Replication process. By the use of Mathematical expression, we can predict
the number of virions that exit from the host cell, and here we have applied
fractal dimension (Cluster growing method) to study the growth of infection.
The stages of the growth structure of Replication is also studied.
28
6. REFERENCES
[1] Amorim MJ and Digard P, “Influenza A virus and the cell nucleus”,
Pud Med., Vol.24, pp.44-46, 2006.
[2] Baki koyuncu and Mahmutsecir, “Student timetable by using Graph
coloring Algorithm”, e-Journal.
[3] Balakrishnan N, “Examination scheduling: a computerized
application”, Omega., Vol.19(1), pp.37- 41,1991.
[4] Clark T and Stephenson I, “Influenza A/H1N1 in 2009 a Pandemic in
evolution”,Pud Med.,Vol. 8(7), pp.819-822, 2009.
[5] G.A.; Merline, D.; Nonnenmacher, T.F.; Weibel, Losa E.R.: Fractals in
Biology and Medicine. (Eds.),springer., Vol.III, Hardcover ISBN:
978-3-7643-6474-8, 2002.
[6] Hampson AW and Mackenzie, Js. “The influenza viruses”,
Med.J.,Vol.185(10), pp.39-43, 2006.
[7] Huawei Mao, Wenwei Tu*, Yinping Liu, Gang Qin, Jian Zheng, Ping-
Lung Chan, Kwok-Tai Lam, J. S. Malik Peiris, and Yu-Lung Lau*:
“Inhibition of human natural killer cell activity by influenza virion and
hemagglutinin”, Journal of Virology., Vol.84, pp. 4148-4157, 2010.
[8] Influenza: taxonomy, facts and Myths, Life cycle, health complications,
treatment options, Influenza.2010.oct.
[9] Kawaguchi A and Nagata K: “Molecular Mechanism of replication and
transcription of the influenza virus genome and host factors virus”,
Pub Med., Vol.56(1), pp.99-108,2006.
[10] Liu.Y, Lou Z, Bartiam M and Rao.Z, “Strucuture function studies of the
influenza virus RNA polymerase PA subunit”, Sci china Life Sci .,
Vol.52(5), pp. 450-458, 2009.
29
[11] Nayak DP, Hui EK and Barman S, “Assembly and budding of
influenza virus”, Virus Res., Vol.106 (2), pp.147-165, 2004.
[12] Song.C, Havlin.S, and H. A. Makse, “Fractal dimension on networks”
Nature (London)., 433, 392 (2005).
[13] Stella SF Ng, Olive TW Li, Timothy KW Cheung, J S Malik Peiris and
Leo LM Poon*, “Heterologous influenza vRNA segments with
identical non-coding sequences stimulate viral RNA replication in
trans”, Virology Journal(e Journal)., Vol.5(2), 2008.
[14] Tasleem Samji, “Influenza A: Understanding the Viral Life Cycle Yale
Journal of Biology & Medicine”, Vol.82(4),pp.153-159, 2009.
[15] Unal Ufuktepe and Goksen Bacak, “Applications of Graph
coloring”, Computational Science and its Applications – ICCSA
2005,Springerlink., Vol.3482, pp.465-477, 2005.
[16] Wright P. Influenza viruses. In: Klieg man RM, Behrman RE, Jenson
HB. Nelson Textbook of Pediatrics. 18th ed. Philadelphia PA: W.B.
Saunders, pp.1384-1387, 2007.
30
8. TECHNICAL BIOGRAPHY
Mr. Magesh Babu .P (RRN. 1040201) was born on 19th July 1985, in
Chennai. He did his schooling in Sri Ramakrishna Hr.Sec. School
(South), Chennai – 17. He received B.Sc degree in Mathematics from
KRMMC of Arts and Science, Adyar, Chennai, affiliated to Madras
University. . He received M.Sc degree in Mathematics from S.I.V.E.T
College, Chennai, and affiliated to Madras University. He is currently
pursuing his M.Phil Degree in Mathematics in B.S.Abdur Rahman
University. His area of interest includes Graph theory. The e-mail ID is:
[email protected] and the contact number is: 98840 02685.
34
Proceedings of the International Conference on Mathematics and Computer Science ICMCS 2011
January 7 -8, 2011
Loyola College, Chennai ISBN
A STUDY OF INFLUENZA VIRION USING
FRACTAL BASED GRAPH THEORY
P.Magesh babu
1, S.Srinivasan
2, P.S.Sheik Uduman
3, C.D.Nandakumar
4, R.S.Ramya
5
1. Department of Mathematics, B.S.Abdur Rahman University, Chennai 48, Tamilnadu, India
2. Professor, Dept. of Mathematics, B.S.Abdur Rahman University, Chennai 48, Tamilnadu, India
3.Associate Professor, Dept. of Mathematics, B.S.Abdur Rahman University, Chennai 48, Tamilnadu, India
4.Assistant Professor, , Dept. of Mathematics B.S.Abdur Rahman University, Chennai 48, Tamilnadu, India
5.Assistant Professor, KRMM college, Chennai , Tamilnadu, India
Email: [email protected]
Abstract -. In this Paper a study of replication stages of influenza
virion in the human body, represented by using graph theory
(vertex coloring) has been made. In addition the infected parts of
the cell are studies by the use of fractal dimension method.
Keywords - vertex coloring, edges, fractal.
I. INTRODUCTION (INFLUENZA VIRION)
Influenza, commonly referred to as the flu, is an infectious
disease caused by RNA viruses of the family
Orthomyxomiridae (the influenza viruses) that affects birds
and mammals. Influenza is described as a viral infection of
the lungs characterized by fever, cough, and severe muscle
aches by Hampson AW, Mackenzie[2],
Metapathogen.Com/Influenza [4]
Diagram 1
REPLICATION
Virion can only replicate in the living cells. Influenza
infection and Replication is a multi – step process. Firstly
the virus breaks and enters into the cell, then delivers its
genome to a site where it can produce new copies of viral
proteins and RNA, assembles these components into new viral
particles and finally exits the host cell.
Once the virus enters the host cell, the core proteins and the
eight segmented negative sense VRNA are transported into the
nucleus. In the nucleus, the structural proteins are separated
from the negative sense VRNA and get transmitted into the
cytoplasm.
Negative senses VRNA are converted to positive senses
VRNA due to the presence of enzymes in the host cell. The
positive senses VRNA produce multi number of copies.
After wards, the positive senses VRNA are converted to
negative senses VRNA due to the presence of enzymes in the
host cell. In addition, the structural protein has done their
multiplication process in the Ribosome’s. Then the negative senses VRNA and structural proteins are combined together
and they form a new virion which contains eight segments of
negative VRNA; In the same way multi number of copies are
delivered from the host cell as observed by Kawaguchi A,
Nagata K [5]
GRAPH THEORY
In this paper the Replication of virion is represented by the use
of Graph theory, especially by using vertex coloring and
labeling. By the application of vertex coloring, the different
stages that occur in Replication of virion are seen from the
Diagram 2. In this Diagram, vertices represent the stages and
the edges represent the connectivity of the different stages.
VERTEX COLORING
In Graph theory, Graph coloring is a special case of Graph
labeling: it is an assignment of Labels traditionally called
“Colors” to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a
graph such that no two adjacent vertices share the same color;
this is called vertex coloring.
Diagram 2
The above graphical representation is about Replication of
Influenza Virion. The vertex which is assigned by Red color
is the Virion which is composed of eight segmented negative
VRNA. The Vertices with Blue color denote the negative
VRNA which are delivered from the Virion. From each
negative VRNA, the Structural proteins are separated; it is
denoted by an arrow mark towards the cytoplasm. In the
cytoplasm n- number of vertices is colored by Brown color,
which represents the multiple numbers of Structural proteins.
The vertices assigned with Bright Green Color denote the
positive VRNA, which are converted from the Vertices with
Blue Color by the use of enzymes present in the Nucleus. The
vertices with Sea Green Color represent the Multiple Numbers
of Copies of positive VRNA. Afterwards, the vertices with
Sea Green Color are converted to Blue Color due to the
presence of enzymes in the host cell. This n – number of
vertices with Blue Color denotes the n – number of negative
VRNA, which are the complex of structural proteins.
Afterwards, every eight Blue colored vertices get composed
and form the Vertex with Red Color. Therefore multiple
copies of Virion exit from the host cell.
Let, X = Sum of the VRNA.
X =
Number of Virion =
FRACTAL
In this paper after explaining the Replication of Virion by the
use of Graph Coloring (Vertex Coloring), we are interested to
apply Fractal dimension for predicting the growth of the
infection.
A fractal is “a rough or fragmented Geometric shape that can be split into parts, each of which is a reduced size copy of the
whole; a property called self – similarity.
THE CLUSTER GROWING METHOD
One virus infected alveoli is chosen randomly. If the minimum
distance l is given, a cluster of virus infected alveoli separated
by at most l from the infected cell which is chosen first. The
procedure is repeated by choosing many virus infected alveoli.
Then the dimension df can be calculated
( from Song.c,
Havlin.S, and H. A. Makse [7])
where is the average mass of the clusters, defined as
the average number of infected alveoli in a cluster. By this
way we can predict the growth of the virus infection in lungs.
Conclusion
In this paper the Replication of virion is represented by using
graph coloring. Here we can easily see the different stages that
occur during the Replication process. By the use of
mathematical expression, we can predict the number of virions
that exit from the host cell, and here we have applied fractal
dimension (Cluster growing method) to study the growth of
infection.
ACKNOWLEDGEMENTS
I would like to thank Honourable Vice Chancellor
Dr.P.Kanniappan, Respected Registrar Dr.V.M.Periasamy and
Professor & HOD of Mathematics Dr.S.Rajasekaran, of our
university for their support and encouragement in our pursuit
of research.
REFERENCES
1. G.A.; Merline, D.; Nonnenmacher, T.F.; Weibel,: Fractals in
Biology and Medicine Losa. E.R. (Eds.) 2002, VIII, 362 p.,
Hardcover ISBN: 978-3-7643-6474-8
2. Hampson AW, Mackenzie, Js. The influenza viruses, Med.J.
Aust.2006.Nov.20.
3. Huawei Mao, Wenwei Tu*, Yinping Liu, Gang Qin, Jian Zheng,
Ping-Lung Chan,
Kwok-Tai Lam, J. S. Malik Peiris, and Yu-Lung Lau*: Inhibition
of human natural killer cell activity by influenza virion and
hemagglutinin . J. Virol. doi:10.1128/JVI.02340-09.
4. Influenza: taxonomy, facts and Myths, Life cycle, health
complications, treatment options. WWW.
Metapathogen.Com/Influenza.2010.oct.
5. Kawaguchi A, Nagata K: Molecular Mechanism of replication and
transcription of the influenza virus genome and host factors virus.
2006 Jun; 56(1); 99-108.
6. Liu.Y, Lou Z, Bartiam M, Rao.Z; Strucuture function studies of the
influenza virus RNA polymerase PA subunit, Sci china Life Sci.
2009 May 52(5): 450-8.
7. Song.C, Havlin.S, and H. A. Makse, Nature (London) 433, 392
(2005).
8. Stella SF Ng, Olive TW Li, Timothy KW Cheung, J S Malik Peiris
and Leo LM
Poon*.:Heterologous influenza vRNA segments with identical
non-coding sequences stimulate viral RNA replication in trans.
Virology Journal 2008, 5:2 doi:10.1186/1743-422X-5-2.
9. Tasleem Samji.: Influenza A: Understanding the Viral Life Cycle.
Yale J Biol Med.
2009December;82(4):153–159.